The way I learned it, is that a "chef" is technically a manager. "Chef", in French, translates to "chief" in English, and is used to designate the person(s) that manage the operation of a kitchen.
Take me, for example. I attended a Culinary Arts program at a nearby college. I was unable to graduate due to financial reasons. However, I took what I did learn, and applied that to my job. I was already working in a restaurant. I have worked in several restaurants, as well as moved into a different state, then back to Oklahoma. I have worked in just about every position in a kitchen, hot foods, cold foods, banquet prep, line cook, short-order cook (in a snack-shop out on the golf course), etc. I help write weekly specials, and even get new food items put on the menu (just got a new sandwich on our Spring Menu, which goes live either TODAY or tomorrow).
Does this make me a chef? NO! I'm not a manager. Heck, I'm not even a supervisor. We only have two Chef's in my kitchen, the Exec. Chef, and the Sous Chef.
Now, turn the table upside-down. Imagine a small, family-run type restaurant. You have a crew that knows their job(s). They've been there for years. You have a "kitchen manager" that oversees the operation of the kitchen. This person is technically the Chef. Does that person have a degree from an accredited school? Maybe, maybe not. Does that person have a lot of experience in running that kitchen? Yes.
In answer to your question, yes, you can become a "Chef" without a degree, or even any kind of formal "schooling" in the industry. However, the more, and better education you can get, the better off you'll be. It might be a good idea to start at the school you mentioned, to get some training under your belt, then get a job in the industry. After that, you can decide if you want to attend a school that will actually let you earn a degree. If I remember right, CIA in Hyde Park won't accept students that don't have experience in the field already, and I think it's two years experience.
There are many other professionals that are members of this website. Hopefully they will their point(s) of view as well.
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